Topological Insulators Bi2Sb3: -First known 3D topological insulator -Highly complex surface states Properties Insulating materials that conduct electricity through gapless surface states The surface states are “topologically protected”, which means that they cannot be destroyed by impurities or imperfections Topological insulators require two conditions: Time reversal symmetry Strong spin–orbit interaction, which occurs in heavy elements such as Hg and Bi. Where do the surface states come from? How did they originate. How were they predicted with theory? (Find the papers) Applications They’re cool! Integrated circuit technology Minimize power dissipation Spintronics Error tolerant quantum computation Scattering may alter, but won’t destroy conductive surface states

Quantum-spin hall effect: 2D topological insulator Time Reversal Symmetry Phase changes are associated with symmetry breaking Magnets & rotation. Topological protected states and time reversal What does TR symmetry mean? Spin and velocity both odd under TR Spin orbit coupling QHE, large applied magnetic field breaks TR symmetry First known topologically protected state Quantum Spin Hall Effect Spin-orbit interaction takes place of magnetic field A general prediction of this state was made in 2005 by Kane and Mele

Experimental Discovery: HgTe Quantum Wells Prediction HgTe QW’s in 2006 Kane also predicted it in graphene at the same time Observation by König et al. 2007 Measured conductance near 0K, which would otherwise be zero. Effective Hamiltonian for 2D Surface states k±=±kx∓iky A2=4.1eV⋅Å VF=A2/ħ⋍6×105 m/s Bernevig, B. A., Hughes, T. L., & Zhang, S. C. (2006). Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science,314(5806), 1757-1761. König, M., Wiedmann, S., Brüne, C., Roth, A., Buhmann, H., Molenkamp, L. W., ... & Zhang, S. C. (2007). Quantum spin Hall insulator state in HgTe quantum wells. Science, 318(5851), 766-770.

Inset: devices from same wafer at different temperatures. Experimental Discovery: HgTe Quantum Wells μHgTe~105 cm2V-1s-1 μSilicon~103 cm2V-1s-1 μgraphene~105-106 cm2V-1s-1 lMFP~1μm At a critical size quantum well size they see the expected behavior, and conductance G0 Compare mobility to graphene Fig. 4. The longitudinal four-terminal resistance, R14,23, of various normal (d = 5.5 nm) (I) and inverted (d = 7.3 nm) (II, III, and IV) QW structures as a function of the gate voltage measured for B = 0 T at T = 30 mK. Inset: devices from same wafer at different temperatures. Markus König et al. Science 2007;318:766-770

Experimental Discovery: HgTe Quantum Wells Four-terminal magnetoconductance, G14,23, in the QSH regime as a function of tilt angle between the plane of the 2DEG and applied magnetic field for a d = 7.3-nm QW structure with dimensions (L × Ω) = (20 × 13.3) μm2 measured in a vector field cryostat at 1.4 K. Four-terminal magnetoconductance, G14,23, in the QSH regime as a function of tilt angle between the plane of the 2DEG and applied magnetic field for a d = 7.3-nm QW structure with dimensions (L × Ω) = (20 × 13.3) μm2 measured in a vector field cryostat at 1.4 K. Markus König et al. Science 2007;318:766-770

3D Topological Insulator Analogy to 2D 1D conducting edge channels become 2D conducting surfaces Line of gapless states becomes a cone of states bridging the bulk conduction and valence bands Can’t just measure conductance near 0K Angle resolved photoelectron spectroscopy First predicted by Kane et al. 2007 (Bix-1Sbx) First observations in 2008 and 2009 Bi2Sb3, Bi2Te3, Bi2Se3, Sb2Te3 Spin resolved photoelectron spectroscopy: be able to explain and write out hamiltonian, and show the spin dependent term that they can measure. Shoot photons of different energy at the material. Measure at which energies electrons escape the material. Knowing the work function of the material, the angle of incidence and ejection of electron you can work backwards and calculate the spin. Hsieh, D., Qian, D., Wray, L., Xia, Y., Hor, Y. S., Cava, R. J., & Hasan, M. Z. (2008). A topological Dirac insulator in a quantum spin Hall phase. Nature,452(7190), 970-974. Fu, L., & Kane, C. L. (2007). Topological insulators with inversion symmetry.Physical Review B, 76(4), 045302.

Computational results for LDOS of four different materials Zhang, H., Liu, C. X., Qi, X. L., Dai, X., Fang, Z., & Zhang, S. C. (2009). Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nature physics, 5(6), 438-442

Crystal and electronic structures of Bi2Te3 Crystal and electronic structures of Bi2Te3. (A) Tetradymite-type crystal structure of Bi2Te3. (B) Calculated bulk conduction band (BCB) and bulk valance band (BVB) dispersions along high-symmetry directions of the surface BZ (see inset), with the chemical potential rigidly shifted to 45 meV above the BCB bottom at Γ to match the experimental result. (C) The kz dependence of the calculated bulk FS projection on the surface BZ. (D) ARPES measurements of band dispersions along K-Γ-K (top) and M-Γ-M (bottom) directions. The broad bulk band (BCB and BVB) dispersions are similar to those in (B), whereas the sharp V-shape dispersion is from the surface state band (SSB). The apex of the V-shape dispersion is the Dirac point. Energy scales of the band structure are labeled as follows: E0: binding energy of Dirac point (0.34 eV); E1: BCB bottom binding energy (0.045 eV); E2: bulk energy gap (0.165 eV); and E3: energy separation between BVB top and Dirac point (0.13 eV). (E) Measured wide-range FS map covering three BZs, where the red hexagons represent the surface BZ. The uneven intensity of the FSs at different BZs results from the matrix element effect. (F) Photon energy–dependent FS maps. The shape of the inner FS changes markedly with photon energies, indicating a strong kz dependence due to its bulk nature as predicted in (C), whereas the nonvarying shape of the outer hexagram FS confirms its surface state origin. Y. L. Chen et al. Science 2009;325:178-181 Published by AAAS

Doping dependence of Fermi Surfaces and EF positions in Bi2Te3 Doping dependence of FSs and EF positions. (A to D) Measured FSs and band dispersions for 0, 0.27, 0.67, and 0.9% nominally doped samples. Top row: FS topology (symmetrized according to the crystal symmetry). The FS pocket formed by SSB is observed for all dopings; its volume shrinks with increasing doping, and the shape varies from a hexagram to a hexagon from (A) to (D). The pocket from BCB also shrinks upon doping and completely vanishes in (C) and (D). In (D), six leaf-like hole pockets formed by BVB emerge outside the SSB pocket. Middle row: image plots of band dispersions along K-Γ-K direction as indicated by white dashed lines superimposed on the FSs in the top row. The EF positions of the four doping samples are at 0.34, 0.325, 0.25, and 0.12 eV above the Dirac point, respectively. Bottom row: momentum distribution curve plots of the raw data. Definition of energy positions: EA: EF position of undoped Bi2Te3; EB: BCB bottom; EC: BVB top; and ED: Dirac point position. Energy scales E1 ~ E3 are defined in Fig. 1D. Y. L. Chen et al. Science 2009;325:178-181

Three-dimensional illustration of the band structures of undoped Bi2Te3 (A) Three-dimensional illustration of the band structures of undoped Bi2Te3, with the characteristic energy scales E0 ~ E3 defined in Fig. 1D. (B to E) Constant-energy contours of the band structure and the evolution of the height of EF referenced to the Dirac point for the four dopings. Red lines are guides to the eye that indicate the shape of the constant-energy band contours and intersect at the Dirac point. Y. L. Chen et al. Science 2009;325:178-181

Thickness dependence of band structure in Bi2Se3 Zhang, Y., He, K., Chang, C. Z., Song, C. L., Wang, L. L., Chen, X., ... & Shen, S. Q. (2010). Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit. Nature Physics, 6(8), 584-588.

Continued Work Discovering more topological insulators Over 50 materials have been predicted, most of which haven’t been experimentally tested Combining with a superconductor to look at majorana fermions People think they might form at interface of ordinary superconductor and topological insulator (no idea why) Topological superconductors Haven’t been observed or predicted, but people are looking

References Papers Resources [1] Kane, C. L., & Mele, E. J. (2005). Quantum spin Hall effect in graphene.Physical review letters, 95(22), 226801. [2] Bernevig, B. A., Hughes, T. L., & Zhang, S. C. (2006). Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science,314(5806), 1757-1761. [3] König, M., Wiedmann, S., Brüne, C., Roth, A., Buhmann, H., Molenkamp, L. W., ... & Zhang, S. C. (2007). Quantum spin Hall insulator state in HgTe quantum wells. Science, 318(5851), 766-770. [4] Fu, L., & Kane, C. L. (2007). Topological insulators with inversion symmetry.Physical Review B, 76(4), 045302. [5] Zhang, H., Liu, C. X., Qi, X. L., Dai, X., Fang, Z., & Zhang, S. C. (2009). Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nature physics, 5(6), 438-442. [6] Xia, Y., Qian, D., Hsieh, D., Wray, L., Pal, A., Lin, H., ... & Hasan, M. Z. (2009). Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nature Physics, 5(6), 398-402. [7] Kane, C., & Moore, J. (2011). Topological insulators. Physics World, 24(02), 32. [8] Zhang, Y., He, K., Chang, C. Z., Song, C. L., Wang, L. L., Chen, X., ... & Shen, S. Q. (2010). Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit. Nature Physics, 6(8), 584-588. [9] Chen, Y. L., Analytis, J. G., Chu, J. H., Liu, Z. K., Mo, S. K., Qi, X. L., ... & Zhang, S. C. (2009). Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science, 325(5937), 178-181. Resources 2011 Qi and Zhang review: http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.1057 Nature Perspective Article: http://www.nature.com/nature/journal/v464/n7286/full/nature08916.html ARPES info: Damascelli, A. (2004). Probing the electronic structure of complex systems by ARPES. Physica Scripta, 2004(T109), 61. More in paper

Angle resolved photoelectron spectroscopy (ARPES) E=binding energy ɸ=work function a high-energy photon is used to eject an electron from a crystal, and then the surface or bulk electronic structure is determined from an analysis of the momentum of the emitted electron.

Experimental Results First 2D Topological Insulator (QsHE )discovered in 2007 1 They measured a conductance of G0 near 0K, independent of sample dimensions. First 3D Topological Insulator discovered in 2008 2 using ARPES, they mapped out the surface states of BixSb1–x and observed a special characteristic of topological insulators ARPES high-energy photons are shone onto the sample and electrons are ejected. By analyzing the energy, momentum and spin of these electrons, the electronic structure and spin polarization of the surface states can be directly measured. There is always some bulk conductance in a 3D material, which can’t easily be separated from surface conductance. Need new experiment: ARPES Maybe graphic of ARPES and then hamiltonian with relevant quantities (angle), spin, etc explaining how it can directly measure structure and spin of surface states. 1 at University of Würzburg, Germany, led by Laurens Molenkamp 2 at Princeton University led by Zahid Hasan

More Info M Z Hasan and C L Kane 2010 Colloquium:Topological insulators Rev. Mod. Phys. 82 3045– 3067 J E Moore 2010 The birth of topological insulators Nature 464 194–198 X-L Qi and S-C Zhang 2010 The quantum spin Hall effect and topological insulators Physics TodayJanuary pp33–38

Ideas of stuff to say Interesting phenomena and phae changes often result from from symmetry breaking Examples Topological insulators are a phase change that results from keeping a symmetry (similar to QHE) Magnetization and velocity are both odd (classically) under time reversal.

Continued Work Using topological insulators to make Majorana fermion

Properties of Topological Insulators A&M p354 discussion to motivate understanding surface effects We have ignored surface effect for the most part so far, often treating out solids as infinite in size. We’re pretty justified in ignoring their overall contribution: 1024 atoms in a typical bulk material, only 108 are on the surface. Surface effects dominate at nanoscale and in low dimensional materials Surfaces tend to be highly irregular with numerous defects making them difficult to study and test predictions. Easily testable prediction because topologically insulating states are protects from defects. Resources to provide so they can ask good questions? -Find derivation type of section in A&M/Kittel that was predicted before experimental observation. After that go into what recent work has done on them.